It depends on the types of games you play.
Some games are graphics limited like fast action shooters.
Others are cpu core speed limited like strategy, sims, and mmo.
Multiplayer tends to like many threads.
One simple test is to Run YOUR games, but lower your resolution and eye candy.
If your FPS increases, it indicates that your cpu is strong enough to drive a better graphics configuration.
If your FPS stays the same, you are likely more cpu limited.
Since you have the capability of overclocking your 6600K, that is the first thing I would do.
As of 12/04/2016
What percent can get an overclock at a somewhat sane 1.40v Vcore.
I5-6600K
4.9 14%
4.8 38%
4.7 67%
4.6 87%
As to a upgrade, your choice will depend on how many threads your games can usefully use.
i3-8350K has 4 threads, i5-9600K has 6, i7-9700K has 8 and I9-9900K has 16.
All will overclock to 5.0
You will need a new motherboard, either Z370 or Z390 based.
Z390 does not cost much more and is recommended for the 9th gen processors.
You would get a nice boost with a 9600K, but for an upgrade a i7-9700K is really going to be as good as it gets for gaming.
Only if budget does not matter would I spring for a i9-9900K.
Most games do not effectively use more than 2-3 threads.
If you are wondering how sensitive your games are to thread count, try this test:
You can do this in the windows msconfig boot advanced options option.
Set your thread count to 3.
You will need to reboot for the change to take effect.
This will tell you how sensitive your games are to the benefits of many threads.
If you see little difference, your game does not need all the threads you have.